A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)

Abstract

Abstract. In this paper, we investigate the relationships between the space X andthe hyperspace C ( X ) concerning admissibility and connectedness im kleinen. Thefollowing results are obtained: Let X be a Hausdorfi continuum, and let A 2 C ( X ).(1) If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ‰ IntK ‰ K ‰ U , then C ( X ) is connected im kleinen. at A . (2) If IntA 6= ; , then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ‰ IntK ‰ K ‰ U . (3) If X is connected im kleinen. at A , then A is admissible. (4) If A is admissible, thenfor any open subset U of C ( X ) containing A , there is an open subset V of X suchthat A ‰ V ‰ S U . (5) If for any open subset U of C ( X ) containing A , there is asubcontinuum K of X such that A 2 IntK ‰ K ‰ U and there is an open subset V of X such that A ‰ V ‰ S IntK , then A is admissible. 0. Introduction Let X be a Hausdorfi continuum, and let 2